Several proof-assistants rely on the very formal basis of Pure Type Systems (PTS) as their foundations. We are concerned with the issues involved in the development of large proofs in these provers such as namespace management, development of reusable proof libraries and separate verification. Although actual implementations offer many features to address them, few theoretical foundations have been laid for them up to now.

: In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the Curry-Howard isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambda-terms. In order to see the process of proof construction as an incremental process of term construction, it is necessary to extend the lambda-calculus with new operators. First, we consider typed meta-variables to represent the parts of a proof that are under construction, and second, we make explicit the substitution mechanism in order to deal with capture of variables that are bound in terms containing meta-variables. Unfortunately, the theory of explicit substitution calculi with typed meta-variables is more complex than that of lambda-calculus. And worse, in general they do not share the same properties, notably with respect to confluence and strong normalization. A contribution of this thesis is to show that the pr...

. The logical and operational aspects of rewriting logic as a logical framework are illustrated in detail by representing pure type systems as object logics. More precisely, we apply membership equational logic, the equational sublogic of rewriting logic, to specify pure type systems as they can be found in the literature and also a new variant of pure type systems with explicit names that solves the problems with closure under -conversion in a very satisfactory way. Furthermore, we use rewriting logic itself to give a formal operational description of type checking, that directly serves as an ecient type checking algorithm. The work reported here is part of a more ambitious project concerned with the development in Maude [7] of a proof assistant for OCC, the open calculus of constructions, an equational extension of the calculus of constructions. 1 Introduction This paper is a detailed case study on the ease and naturalness with which a family of higher-order formal systems, namely...

This paper surveys a part of the theory of fi-reduction in -calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from -terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in -terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in -calculus and type theory. 1. Introduction Considerable attention has been devoted to classification of reduction strategies in type-free -calculus [4, 6, 7, 15, 38, 44, 81]---see also [2, Ch. 13]. We are concerned with strategies differing in the length of reduction paths. This paper draws on several sources. In late 1994, van Raamsdonk and Severi [59] and Srensen [66, 67] independently developed ...

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This paper surveys a part of the theory of fi-reduction in λ-calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λ-terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λ-terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in λ-calculus and type theory.

We show that every functional Pure Type System may be extended to a semi-full Pure Type System. Moreover, the extension is conservative and preserves weak normalization. Based on these results, we give a new, conceptually simple type-checking algorithm for functional Pure Type Systems. 1

We present a dependent-type system for a #-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.

Injective Pure Type Systems form a large class of Pure Type Systems for which one can compute by purely syntactic means two sorts elmt(\GammajM ) and sort(\GammajM ), where \Gamma is a pseudo-context and M is a pseudo-term, and such that for every sort s, \Gamma ` M : A \Gamma ` A : s ) elmt(\GammajM ) = s \Gamma ` M : s ) sort(\GammajM ) = s By eliminating the problematic clause in the (abstraction) rule in favor of constraints over elmt(:j:) and sort(:j:), we provide a sound and complete type-checking algorithm for injective Pure Type Systems. In addition, we prove Expansion Postponement for a variant of injective Pure Type Systems where the problematic clause in the (abstraction) rule is replaced in favor of constraints over elmt(:j:) and sort(:j:). 1

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